Moulson A.J., Herbert J.M. Electroceramics: Materials, Properties, Applications

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hexagonal crystals the anisotropy energy E

depends only upon the angle y

between the c axis and the magnetization vector, and is almost independent of

direction in the basal plane. It can be expressed approximately as

¼ K

sin

y þ K

sin

y ð9:29Þ

The energy minimum occurs for y ¼ 0, making the c axis the ‘easy’ direction.

The concept of an anisotropy magnetic ﬁeld H

, which is referred to simply as

the ‘anisotropy ﬁeld’, is also introduced to describe magnetic anisotropy. The

anisotropy energy is considered to be that of the saturation magnetization

moment M

in an induction m

, i.e. E

¼m

cos y, where y is the angle

between H

and M

9.1.8 Magnetostriction

Because of the spin–orbit lattice coupling referred to in the previous section,

changes in the spin directions result in changes in the orientation of the orbits

which, because they are restrained by the lattice, have the eﬀect of slightly

altering the lattice dimensions. This eﬀect is known as magnetostriction.

The magnetostriction constant l

is deﬁned as the strain induced by a saturating

ﬁeld; it is given a positive sign if the ﬁeld causes an increase in dimensions in the

ﬁeld direction. For single crystals l

varies with the crystallographic direction, and

so for the ceramic form it is an average of the single-crystal values. l

values for

some polycrystalline ferrites are given in Table 9.2.

9.1.9 Weiss domains

The fact that spontaneous magnetization exists in, for example, a piece of iron,

and yet the overall magnetization can be zero, is explained by the existence of

482 MAGNETIC CERAMICS

Table 9.1 Room temperature anisotropy constants

of some important ferrites

Ferrite K

/kJ m

711

0.98

1.86

72.8

0.8

2.2

+290

NiFe

76.2

CuFe

76.0

0.45

0.55

70.38

0.5

BaFe

+330

TEAMFLY

Team-Fly

domains. Below its Curie temperature a ferromagnetic or ferrimagnetic body

consists of a large number of small domains, each spontaneously magnetized to

saturation. Each grain or crystallite in a polycrystalline magnetic ceramic may

contain a number of domains, each diﬀering from its neighbour only in the

direction of magnetization.

A single crystal with uniform magnetization, i.e. a single-domain single crystal,

has magnetostatic energy due to the external magnetic ﬁeld which it generates. If

the crystal is divided into oppositely oriented parallel domains, the energy will be

greatly reduced since the ﬂux can now pass from one to another of the closely

adjacent domains (Fig. 9.8(a)). In cubic materials, such as spinels and garnets,

zero magnetostatic energies are possible through the formation of closure

domains (Fig. 9.8(b)) since the external ﬂux is now close to zero. Even so, the

system has magnetoelastic energy because magnetostriction leads to straining

between the long and the triangular domains.

The boundary between two adjacent domains is known as a domain wall or

Bloch wall – the latter term after F. Bloch who proved in 1932 that

magnetization cannot change discontinuously at a domain boundary. A Bloch

wall is the region between two domains in which the elementary spin moments

change smoothly from one orientation to another. For example, in the case of

the antiparallel domains the change in direction of the vectors in moving from

one domain to an adjacent one would be as shown diagrammatically in Fig. 9.9.

The walls have widths in the range 10–100 nm and an associated energy in the

range (1–10)610

An important property of a Bloch wall is its mobility. It can be seen from

Fig. 9.9 that the application of H in the sense shown will cause the wall to move

to the left by a series of minor rotations of the vectors. A more detailed

consideration of wall displacements shows them to be of two types, reversible or

irreversible; which one occurs depends upon the size of the displacement and the

MAGNETIC CERAMICS: BASIC CONCEPTS 483

Table 9.2 Saturation magnetostriction constants

for some polycrystalline ferrites

Composition l

/10

+40

MnFe

CoFe

7110

NiFe

726

0.56

2þ

0:44

0.5

711

MgFe

TbFe

a ‘giant’ magnetostrictor 410

*Included for comparison (H. Uchida et al. (2002) Progress in thin

ﬁlms of giant magnetostrictive alloys, J. Mag. Mag. Mater., 239,

540–5).

disposition of wall energy minima, which in turn depend upon crystal lattice

defects and inhomogeneities of various types.

9.1.10 Magnetization in a multidomain crystal

The most characteristic feature of ferromagnetic or ferrimagnetic materials is the

relationship between B and H (or M and H) – the hysteresis loop (Fig. 9.10) –

which can now be considered in more detail in terms of the magnetization

processes in a multidomain single crystal. The line deOba – the ‘virgin curve’ –

represents the relationship determined experimentally when the specimen is

demagnetized before each measurement of the induction for a given ﬁeld. The

change in B, very near to the origin, represents magnetization by reversible Bloch

wall displacements, and the tangent OC to this initial magnetization curve is

called the initial permeability m

. The steep rise in B represents magnetization by

irreversible Bloch wall displacements as the walls break away from their pinning

points, and the region ba represents magnetization by reversible and irreversible

domain rotations from one easy direction to another more favourably aligned

with the applied ﬁeld. The latter process requires high ﬁeld strengths because the

magnetization within a domain is rotated against the anisotropy ﬁeld.

484 MAGNETIC CERAMICS

Fig. 9.8 Idealized magnetic domain conﬁgurations: (a) antiparallel domains; (b) ﬂux closure

domains.

Fig. 9.9 The change in spin orientation across the width of a Bloch wall.

The slope of Oa, from the origin to the tip of the loop, gives the amplitude

permeability m

which has a maximum value when the peak ﬁeld corresponds to

the point b on the virgin curve. If a relatively small alternating ﬁeld is

superimposed on a static ﬁeld, a minor loop such as ef is obtained, and the

amplitude permeability of this is known as the incremental permeability m

.As

far as applications are concerned, m

is important for inductors where only small

alternating ﬁelds are encountered, m

is important for power transformers when

large alternating ﬁelds are involved and m

is important for transductors to

which both alternating and static ﬁelds are applied.

If, after the material has been magnetically saturated to the value B

, the ﬁeld

is reduced to zero, the magnetization vectors rotate out of line with the ﬁeld

towards the nearest preferred direction which is determined in part by

magnetocrystalline anisotropy. The magnetization is thus prevented from

complete relaxation to the ‘virgin’ curve and hence, for zero ﬁeld, there is a

remanent induction B

. In order to reduce the induction to zero a reverse ﬁeld H

has to be applied. The coercive ﬁeld or ‘coercivity’ H

depends in part on

crystalline anisotropy, as might be expected.

Because of hysteresis, energy is dissipated as heat in a magnetic material as it is

taken round a complete B –H loop, and the hysteresis energy loss W

per unit

volume of material is

BdH ð9:30Þ

Magnetic materials are usually characterized as ‘hard’ or ‘soft’, depending on the

magnitude of their coercivity and Figure 9.11 shows typical loops.

MAGNETIC CERAMICS: BASIC CONCEPTS 485

Fig. 9.10 Magnetic B–H hysteresis loop.

9.2 Model Ferrites

9.2.1 Spinel ferrites: model NiOFe

Magnetite (Fe

), a naturally occurring ferrite, is the earliest known magnetic

material and was exploited as lodestone

{

many hundreds of years ago. Its

composition can be written FeOFe

, when the structural relationship to the

mineral spinel (MgOAl

) is apparent. There are many other possible

compositions with the general formula MeOFe

, in which Me represents a

divalent ion such as Mn

,Fe

,Co

,Ni

,Cu

or Zn

, or a combination

of divalent ions with an average valence of 2. The substitutional possibilities are

considerable, which has led to the extensive technological exploitation of ferrites.

In the spinel crystal structure the oxygen ions form a cubic close-packed array

in which, as discussed in Chapter 2 (Fig. 2.1), two types of interstice occur, one

coordinated tetrahedrally and the other octahedrally with oxygen ions. The cubic

unit cell is large, comprising eight formula units and containing 64 tetrahedral

and 32 octahedral sites, customarily designated A and B sites respectively; eight

of the A sites and 16 of the B sites are occupied. The unit cell shown in Fig. 9.12

is seen to be made up of octants, four containing one type of structure (shaded)

and four containing another (unshaded). In this representation some of the

A-site cations lie at the corners and face-centre positions of the large cube; a

486 MAGNETIC CERAMICS

Fig. 9.11 Hysteresis loops illustrating the distinction between magnetically ‘soft’ and ‘hard’

materials.

{ The attractive power of magnetite was known to the Greeks (c. 100 BC); by the early 17th century – and

probably much earlier – it was used by the Chinese to magnetize compass needles. Hence, ‘lode’ in

‘lodestone’ is synonomous with ‘lode’ in ‘lodestar’ (the Pole star).

tetrahedral and an octahedral site are shown. The close-packed layers of the

oxygen ion lattice lie at right angles to the body diagonals of the cube. The

arrows on the ions, representing directions of magnetic moments, indicate that

the B-site ions have their moments directed antiparallel to those of A-site ions,

illustrating the antiferromagnetic coupling.

The occupancy of sites in the spinel-type minerals is conveniently represented

with the help of the diagrams in Fig. 9.13. In the case of the mineral spinel the

divalent ions (Mg) occupy the A sites and the trivalent ions (Al) the B sites

(Fig. 9.13(a)). This is known as the normal spinel structure. In the spinel ferrites

(MeFe

) diﬀerent ions exhibit diﬀerent site preferences. In nickel ferrite, for

instance, the Ni

ions occupy B sites along with an equal number of randomly

distributed Fe

ions, whilst the remaining Fe

ions occupy A sites. This is

termed an inverse spinel structure (Fig. 9.13(b)). In practice, the structures of the

spinel-type magnetic ceramics are a blend of the normal and inverse types, with

the oxidation states and distribution of cations amongst the lattice sites

depending upon thermal history, i.e. on ﬁring schedules and kiln atmosphere.

Attention is now given to the ordering of the magnetic moments. The spin-

ordering forces are of the ‘superexchange’ kind, so called because they act

between the resultant spin on the cations through the agency of an intervening

oxygen ion. Superexchange forces acting between A ions, between B ions and

MODEL FERRITES 487

Fig. 9.12 The unit cell of a magnetic inverse spinel.

Fig. 9.13 Diagrammatic representation of site occupancy in (a) normal and (b) inverse spinels.

between A and B ions are referred to as AA, BB or AB interactions, and in each

case the forces tend to hold the spins of nearest-neighbour cations antiparallel.

The AB interaction is the strongest and in consequence moments on the A ions

are directed antiparallel to those of the B ions. Ne

el introduced and developed

this theory in 1948.

Before proceeding further with the analysis, the magnitude of the magnetic

moments associated with the various cations must be known and, because of

orbital moment quenching, only the spin moments need be considered when

calculating the moment of the ions. For example, as a ﬁrst step in calculating the

magnetic moment of NiOFe

it is readily shown that the resultant spin

quantum numbers for Ni

and Fe

ions are 1 and 5/2 respectively. Because the

spinning electron produces twice the magnetic moment expected from its angular

momentum, the corresponding magnetic moments are 2m

and 5m

. Also,

because all the A moments are directed antiparallel to the B moments, it is

evident from Fig. 9.13(b) that there is a resultant moment of 16m

per unit cell

which is due to the eight Ni

ions. The saturation magnetization can now be

calculated from the cell dimensions:

16m

cell volume

16  9.27  10

24

ð8.37  10

10

 2:5  10

1

This compares well with the experimentally determined value (approximately

3610

). The discrepancy is probably due partly to the assumption that

NiOFe

has the ideal inverse spinel structure and partly to the incomplete

quenching of the orbital moment.

An elegant conﬁrmation of the essential correctness of the theory lies in the

explanation of why the addition of a non-magnetic ion such as Zn to a spinel

ferrite should lead to an increase in saturation magnetization. The magnetic

moment per formula unit MeFe

, where Me represents Ni

or Zn

, is shown

in Fig. 9.14 as a function of zinc substitution. Zinc ferrite is a normal spinel,

indicating that Zn ions have a preference for the A sites, so that on substituting

zinc for nickel the occupancy becomes (Fe

3þ

1d

2þ

)(Fe

3þ

1þd

2þ

1d

, in which the

ﬁrst and second brackets indicate occupancy of the A and B sublattices

respectively. Thus the antiparallel coupling between moments on A and B sites is

reduced because the occupancy of A sites by magnetic ions is reduced, and as a

consequence the Curie point is lowered (see Fig. 9.18 below). However, the

excess of moments on octahedral sites over those on tetrahedral sites is increased

so that the magnetization is increased. The data plotted in Fig. 9.14 conﬁrm this

model up to d  0.4. The fall in magnetization for higher values of d is due to the

reduced antiparallel coupling between the A and B sites, and it becomes zero

when d ¼ 1, i.e. (Zn

)(Fe

3þ

Generally speaking, the spinel ferrites have low magnetic anisotropies and are

magnetically ‘soft’; exceptions are those containing Co

which is itself strongly

488 MAGNETIC CERAMICS

magnetically anisotropic. Cobalt spinel ferrites can have coercivities approaching

, placing them ﬁrmly in the ‘hard’ category.

9.2.2 Hexaferrites: model BaFe

Barium hexaferrite (BaFe

) is the model for a family of ‘M-type ferrites’, so-

called because they are based upon the hexagonal magnetoplumbite, or M-

structure. Its crystal structure, though related to that of the spinels, is very much

more complex. The large unit cell (c ¼ 2.32 nm; a ¼0.588 nm) contains two

formula units, i.e. a total of 64 ions. The Ba

and O

ions together form a

close-packed structure with some of the layers cubic close-packed and others

hexagonal close-packed. The origins of the magnetic properties are basically the

same as those already discussed and can be summarized as follows: of the 12

ions in a formula unit, nine are on octahedral sites two on tetrahedral sites

and one on a ﬁve-coordinated site; seven of the ions on octahedral sites and the

one on a ﬁve-coordinated site have their spins in one sense, and the remainder

are oppositely directed. Thus there are four more ions with spins in the one

sense than there are with spins in the other and, since there are ﬁve electrons

with parallel spins in each Fe

ion, there are 20 unpaired spins per formula

unit, leading to a saturation magnetization of 20 m

per cell volume. BaFe

has a high magnetic anisotropy (Table 9.1) with its ‘prefered direction of

magnetization’ (‘easy’ direction) along the c-axis. Various substitutions are

made to tailor intrinsic magnetic properties, for example Sr for Ba, and partial

substitution of Al for Fe to increase coercivity [8].

9.2.3 Garnets: models Y

(YIG)

‘Garnet’ is the name of a group of isostructural minerals with the general

composition 3R

O  R@

 3SiO

. Examples are 3CaO  Al

 3SiO

MODEL FERRITES 489

Fig. 9.14 Saturation magnetization per ‘formula unit’ for the ferrite (Fe

3þ

1d

2þ

)

(Fe

3þ

1þd

2þ

1d

as a function of d.

(grossularite), 3CaO  Fe

 3SiO

(andradite) and 3MnO  Al

 3SiO

(spes-

sarite). Yttrium iron garnet (YIG) is the best known of a family of ferrimagnetic

garnets because of its importance as a microwave material. It has a large cubic

unit cell (a  1200 pm) containing 160 atoms.

The general formula for the ferrimagnetic garnets is written R

, where

R stands for yttrium in the case of YIG; the yttrium can be totally or partially

replaced by one of the lanthanides such as lanthanum, cerium, neodymium,

gadolinium etc. Therefore the structure contains two types of magnetic ion, iron

and one of the rare earth group. Whilst the contribution to the magnetization

from the orbital motion of the electrons in elements of the ﬁrst transition series is

close to zero (quenching) because of the orbital–lattice coupling, that of the

electrons in the lanthanide ions has a signiﬁcant eﬀect. The unpaired electrons of

the ﬁrst series elements are in the outermost 3d group and therefore are not

shielded from the crystal ﬁeld which is responsible for quenching. In the

lanthanide ions the unpaired electrons in the 4f group are shielded by the 5s5p

electrons and there is therefore an orbital contribution in addition to that of the

unpaired spins. As a consequence the contribution of the lanthanide ions to the

magnetization is somewhat greater than would be estimated from the simple

rules governing the elements of the ﬁrst transition series. A further consequence

of this shielding is that the coupling of lanthanide ions to other magnetic ions is

weaker than that between the ions of the ﬁrst transition series.

The lattice site occupancy is conventionally represented by the formula

}

[Fe

]

(Fe

)

, where [ ]

indicates ions on octahedral sites, ( )

indicates

ions on tetrahedral sites and { }

indicates ions on 12-coordinated sites. There is

strong coupling with antiparallel spins between ions on the a and d sites, and

thus, since all are Fe

, the net contribution is 5m

; the rare earth ions on the c

sites have their unpaired spins coupled antiparallel to the Fe

on d sites and so

contribute 3m

, where m

is the strength of the moment of the R ion measured

in Bohr magnetons. The resultant magnetization per formula unit, measured in

Bohr magnetons, is therefore

M ¼ 5 3m

in which m

is greater than 7 for gadolinium, terbium and dysprosium, and falls

oﬀ to 3.5 for thulium, 2.7 for ytterbium, and zero for lutetium and yttrium.

When m

45/3, M is negative and its value at 0 K is dominated by the

contribution from the rare earth ions. At higher temperatures the rare earth

contribution decreases because of the weak coupling between (Fe

)

and {R}

and thus the magnetization ﬁrst falls to zero and then increases again. The zero

magnetization point is known as the compensation point. Since at high

temperatures magnetization in those rare earth garnet ferrites containing no

substitute for Fe

depends mainly on the (Fe

)

–[Fe

]

coupling, they all

have approximately the same Curie points.

490 MAGNETIC CERAMICS

The types of saturation magnetization–temperature characteristic for the

various garnet ferrites are summarized in Fig. 9.15. Figure 9.16 illustrates for a

series of the yttrium gadolinium iron garnets an important feature associated

with a compensation point which is exploited in certain applications: the

magnetization can be arranged to be almost independent of temperature over a

chosen temperature range. For example, for the composition corresponding to

x¼0.6, i.e. Y

1.2

1.8

, the saturation magnetization is relatively stable over

a wide temperature range centred around 50 8C.

A further important feature of the garnets is their high electrical resistivity

resulting in a high Q suitable for certain microwave devices. This is because they

MODEL FERRITES 491

Fig. 9.15 Variation of saturation magnetization with temperature for various garnets.

Fig. 9.16 Variation of saturation magnetization with temperature for Y

3(17x)

for

various x values.